M & N are connected smooth manifolds without boundary. We have a smooth map F:M$\rightarrow$N,take m and n to be dimensions of M and N res.
A$\subset$M A ={x$\in$M|rank${F}_{*}|_{(x)}$ =k}. Where k is fixed number which is neither minimal(0) nor maximal rank, can we comment anything on topological properties of A. Certainly we can find examples in which A is closed? e.g. take map (x,y)$\rightarrow$(sinx,siny), rank 1 subset is closed here. Can we have F,M,N,k so that A is open or some example in which A is not closed?
Let $R_k$ be the set of points where $F_*$ has rank at most $k$. Then for all $k$, $R_k$ is a closed set (in local coordinates, having rank at most $k$ can be expressed in terms of vanishing of certain determinants of partial derivatives, which gives a closed set since the partial derivatives are continuous). Your set $A$ is then $$R_k\setminus R_{k-1},$$ so it is always the difference of two closed sets, or the intersection of a closed set and an open set. Typically this will be neither open nor closed, but it will be closed if the open set is clopen (i.e., if $R_{k-1}$ is clopen) and open if the closed set is clopen (i.e., $R_k$ is clopen). So, you can get a variety of examples by just arranging for $R_k$ and $R_{k-1}$ to be closed sets with certain properties.
For a simple example where $A$ is neither open nor closed, consider $F:\mathbb{R}^2\to\mathbb{R}^2$ given by $F(x,y)=(x^2,y^2)$ and $k=1$. Examples where $A$ is open are very easy: just take any diffeomorphism, so that $F_*$ has maximal rank anywhere so $A$ will be empty. For an example where $A$ is open and nonempty, consider $F:\mathbb{R}^2\to\mathbb{R}^2$ given by $F(x,y)=(x,0)$ and $k=1$. For an example where $A$ is open but not closed, consider $F:\mathbb{R}^2\to\mathbb{R}^2$ given by $F(x,y)=(x^2,0)$ and $k=1$. By changing the second coordinate to be $e^{-1/y}$ for $y>0$, you can get an example where $A$ is open even though $R_k$ is not open.